I just figured this out for myself because I was not able to find any viable solution online. I recently got an iPad Pro 12.9 inch, and I noticed my Reminders app seemed to be not working right because I had no badges showing on the home screen whereas on my iPhone, it was showing the number of badges correctly and I wondered why it wasn’t syncing over when everything else on my iPad seemed to be syncing just fine. What I figured out was on my iPad where I wasn’t getting badges displaying, I went into the Reminders app and pushed the far left of the four icons on the top of the screen, the calendar icon, and it immediately needed my permission to allow reminders to send notifications. As soon as I allowed the Reminders app to have access to sending me notifications, I suddenly was able to see the Reminders app under the notification settings, whereas it was not even visible before.
So, this solution seems to help people who may have just set up a new iDevice, under the same Apple ID, and has done everything like you’re supposed to do to got everything to be synced and working and synchronization, yet you’re seeing something like the Reminders app just not showing the badges despite you having the same exact settings enabled on the device that IS working how you want it to work.
Also, now that I’ve just figured this out, I’m suddenly reminded that I had to do this a long time ago on a different device and I had trouble figuring out why I couldn’t get the badges to show then and it turns out this is how you have to do it.
So again, to recap, using the device you’re not getting badge notifications on your home screen for the Reminders app, go into the Reminders app and simply tap the calendar icon across the four icons, and it should prompt you to allow access for notifications when you select one of the options shown (today in my case) for the Calendar icon. As soon as I did that, everything was fixed. 
Also, something you can check before you do this, to make sure you’re on the same page as I was with this problem, you’ll notice if you go into the settings for reminders that a notification option would be absent. It would also be absent if you go to the notification settings themselves and the reminders app is not listed under the main notification settings for your whole iPhone.
I hope this helps anyone with reminders app issue(s). Reminders is one of my most used and counted on applications on all of iOS. It truly helps me stay organized, mentally and practically and I love that app. had to figure this out on my own. I also attached a screenshot of the four icons at the top of the Reminders app, I mentioned. (For future reference, the solution was applied on iOS 17.1.1)
